Activation Analysis

G. Boyd*)

The striking development of nuclear science over the last decade and a half has led to consequences of great importance for all the natural sciences. Analytical chemistry and, in particular, inorganic analysis have benefited especially from this development. Alongside others, a new method has become possible for detecting contaminants in various materials by forming their artificially radioactive isotopes. This original method, which may be called the “method of radioactivation analysis,” or, more briefly, “activation analysis,” is, by virtue of the exceptional sensitivity attainable in measurements of radioactivity, above all a method for detecting minute impurities. Thanks to the existence of distinct half-life periods of artificial isotopes and to the characteristic properties of the radiations they emit, this method is to a very high degree free from the interfering influence of other elements. Moreover, the danger of contamination during analysis, always present in ordinary methods for determining impurities, is almost completely absent. Finally, the physical basis of this method proves to be sufficiently broad, since the phenomenon of artificial radioactivity is widespread throughout the periodic table. At present the existence has been established of more than 700 radioactive isotopes of all the known 96 elements, and their number is increasing almost daily.

Since the discovery of artificial radioactivity in 1934, several cases of the application of activation-analysis methods have been described in the literature. Apparently the earliest were the works of Hevesy and Levi \(^{15}\) and of Seaborg and Livingood \(^{32}\). The first authors used the comparatively strong activity appearing in dysprosium and europium upon capture of slow neutrons in order to determine to what extent these elements are present in compositions

*) G. E. Boyd, Analytical Chemistry, 21, 335 (1949). Translated from English.

rare earths. For example, comparison of the intensity of the observable 2.4-hour dysprosium period, appearing upon irradiation of yttrium-contaminated preparations, with analogous intensities of dysprosium—yttrium mixtures of known composition made it possible to determine amounts of dysprosium below 0.1%. Direct determination of a europium concentration of the order of 1% in a gadolinium preparation was carried out by a similar method. The particular value of this method lies in the fact that it makes it possible to avoid the traditional difficulties of chemical separation required when applying ordinary methods of rare-earth analysis. In their work on the 36-inch cyclotron, Seaborg and Livingood showed that after bombardment of iron with 6.4 MeV deuterons one can detect in this iron, otherwise very pure in all other respects, extremely insignificant impurities of gallium. The authors’ estimate of the concentration of the gallium impurity, of the order of \(6 \cdot 10^{-6}\), was based on the assumption of the magnitude of the gallium nucleus cross section for fast deuterons, which at that time seemed quite reasonable.

Recently, similar investigations have been described by King and Henderson \(^{20}\), who detected copper impurities in pure silver in amounts not exceeding several hundredths of a percent; for this purpose they used the reaction of 16 MeV \(\alpha\)-particles with copper, leading to the formation of radiogallium: \(\mathrm{Cu}^{63}(\alpha, \mathrm{n})\) 9.4-hour \(\mathrm{Ga}^{66}\). Sagan and co-workers \(^{30}\), using 3 MeV deuterons, detected the presence of traces of sodium in aluminum at a concentration of \(10^{-5}\). An excellent detailed description of the use of fast charged particles for activation analysis can be found in the report by Ardenne and Bernhard \(^{3}\), who determined the amount of carbon in steel with the aid of deuterons of energy 0.8 MeV from an electrostatic accelerator. A method was developed for very rapid determination of carbon at concentrations below 0.05%.

It may seem that with the aid of fast charged particles one succeeds in detecting smaller amounts of impurities than in those cases where neutrons are used, but in reality this is not so. After the appearance, approximately 6 years ago, of the nuclear reactor, chemists gained access to fluxes of slow neutrons a million times greater in intensity than those available to Hevesy. As a result, beginning in the second half of 1943, analysis by radioactivation became an everyday occurrence in the determination of impurities both in pure chemical compositions and in metals and alloys and even in organic compounds. The value of the method increased especially with the establishment of mass production of radioelements for wide distribution. In this case it is used to determine the degree of purity of materials used in the reactor, and also serves as a control during the purification of these materials.

In the literature, comparatively few examples have so far been given of the exceptional sensitivity of methods that use the nuclear reactor as an analytical instrument. In some cases it has proved possible to determine impurities in quantities already inaccessible to ordinary spectrochemical analysis. Thus, for example, concentrations of iridium not exceeding \(10^{-9}\) can be detected in the best commercial grades of pure metallic platinum. In connection with the recent use of ion-exchange chromatography for the separation of rare earths \(^{19}\), the presence of less than \(10^{-3}\%\) of thulium in spectrochemically pure sesquioxide of erbium was proved by irradiating the sample in the reactor and isolating the rare-earth impurities. Later, Clark and Overman \(^{11}\), in a special study devoted to the method of activation analysis, showed that in samples of ultra-pure commercial hydrous aluminum oxide one can detect manganese impurities in amounts smaller than \(6\cdot 10^{-5}\%\).

During the past year there have also been reports on the use of modern intense sources of slow neutrons for estimating microquantities of gallium and palladium in meteorites \(^{8}\) and for analyzing the microcomposition of biological tissues \(^{37}\).

A number of considerations lead one to think that, in the final analysis, the method of activation analysis will find broad application. Of course, even from the examples cited above it is not difficult to see that the new method is still only at the very beginning of its development.

FUNDAMENTALS OF THE METHOD

The sample to be analyzed is irradiated with a homogeneous flux of fast charged particles or of fast or slow neutrons until a measurable quantity of the radioisotope of the element under study has been formed. The rate of increase in the number of radioactive atoms \(N^*\) with time is given by the well-known formula \(^{26}\)

\[ \frac{dN^*}{dt}=f\sigma_{\mathrm{ak}}N-\lambda N^*, \tag{1} \]

which, after integration, gives:

\[ N^*=\frac{f\sigma_{\mathrm{ak}}N}{\lambda}\left(1-e^{-\lambda t}\right). \tag{2} \]

In these equations \(f\) denotes the flux of bombarding particles, expressed as the number of particles passing through one square centimeter per second; \(\sigma_{\mathrm{ak}}\) is the isotopic cross section of the nuclear reaction under consideration and is expressed as the number of square centimeters per target atom; \(N\) is the number of target atoms; \(\lambda\) denotes the radioactive decay constant and is related to the half-life \(T_{1/2}\) by the relation \(\lambda=0.693/T_{1/2}\).

The activity \(A_t\), determined by the number of decays per unit time, for a total number of radioactive atoms \(N^*\) formed during time \(t\), is given by the expression

\[ A_t=\lambda N^*=f\sigma_{\mathrm{act}}N(1-e^{-\lambda t}) =A_\infty\left(1-e^{-\frac{0.693t}{T_{1/2}}}\right). \tag{3} \]

In equation (3) the product \(f\sigma_{\mathrm{act}}N\) may be regarded as equal to the saturation activity \(A_\infty\), since it obviously expresses the activity obtained after an infinitely long irradiation. The expression in parentheses may be called the “saturation factor” \(S\); it takes values between zero and unity. Thus, for the case described by equation (1), one may say that the activity at any time \(t\) is given by the product of the saturation activity and the saturation factor.

The applicability of the equations given above will be determined by the correctness of the assumptions underlying their derivation. In deriving equation (1), for example, it was assumed that the rate of formation of radioactivity, determined by \(f\sigma_{\mathrm{act}}N\), does not change during the irradiation. Such a condition necessarily presupposes that the flux of charged particles or neutrons passing through the sample does not change with time. It is further assumed that the mean energy of the particles of the flux is constant, since the activation cross section \(\sigma_{\mathrm{act}}\), generally speaking, is a function of this energy and will be constant only at constant energy. Finally, the initial number of target atoms must not decrease noticeably as a result of the nuclear reaction taking place. This last circumstance depends on the magnitude of \(f\) and \(\sigma_{\mathrm{act}}\), the product of which, fortunately, almost never reaches values that would require taking this new complication into account. If all the above-mentioned conditions are fulfilled, then equations (1)—(3) describe the increase in the number of radioactive atoms of a given type up to the moment when the bombardment time \(t\) ends. Subsequently, the activity \(A_t\) formed in this way will decrease with the characteristic half-life, so that the amount of radioactivity in the sample \(A_{t'}\) at time \(t'\) after the end of the bombardment will be determined by the equation

\[ A_{t'}=A_t e^{-\lambda t'}=A_\infty e^{-\lambda t'}(1-e^{\lambda t}). \tag{4} \]

In some cases decay leads not to the formation of a stable atom, but to the formation of a daughter radioactive isotope. If the decay period of the daughter atom is considerably shorter than the period of the original radioactive atom, then a decrease in the total activity will be observed immediately after the end of the bombardment. A quantitative treatment of this and other types of chain processes may be found in the well-known book by Rutherford, Chadwick, and Ellis \(^{29}\).

The discussion of the factors determining the sensitivity of the activation-analysis method can be simplified if equation (3) is written in the form:

the mass of an element of natural isotopic composition (in grams) is equal to

\[ \frac{A_t M}{6.02 \cdot 10^{23} \cdot f \cdot \sigma_{\mathrm{ak}} \left(1-e^{-\frac{0.693\,t}{T_{1/2}}}\right)\theta}, \tag{5} \]

where \(M\) is the chemical atomic weight of the element sought, \(\theta\) is the concentration of the target isotope in the natural element; the meanings of the remaining symbols are the same as above. In order to detect a very small amount of a given element, it is necessary, according to equation (5), that the flux and the activation cross section be large, that the radioelement formed have a short half-life, and, finally, it is desirable that the element contain as large an amount as possible of the activated isotope and that its atomic weight be small.

In each particular case the sensitivity is also determined by the nature of the radiation emitted in the decay of the radioisotope and by the conditions under which it is measured. Thus, for example, the number of \(\beta\)-particles registered by a Geiger–Müller counter per unit time, \(C^\beta\), may be related to the number of particles emitted per unit time, \(A_t^\beta\), by the equation

\[ C^\beta = A_t^\beta \varepsilon \frac{G}{100} f_a f_s, \tag{6a} \]

\[ \frac{C^\beta}{A_t^\beta}=Y, \tag{6b} \]

where \(\varepsilon\) is the efficiency of the detector (counter) for the radiation being measured, i.e., the probability that a particle or quantum produces in it the necessary ionization; \(G\) is the geometrical factor, namely the solid angle expressed by the sensitive volume of the detector, with the source located at the vertex; \(f_a\) is the absorption factor, taking into account absorption of radiation in the wall of the detector, in the air between the source and the detector, in the sample itself, etc.; \(f_s\) is the scattering factor, taking into account scattering in the source and in the surrounding bodies, backscattering from the sample holder, etc.; \(Y\) is the counting efficiency, i.e., the ratio of the number of registered particles of the given type to the total number emitted by the measured source \(^{16}\).

From equations (6) it can be seen that the ratio of the number of registered particles to the total number emitted is determined by the product of four separate factors.

When \(\beta\)-particles are registered with Geiger–Müller counters of the usual type, \(\varepsilon\) is practically equal to unity over a wide interval

energy. The efficiency of counting quanta is usually considerably less than unity and depends on the energy. For X-rays one can obtain values of \(\varepsilon\) reaching 0.1 if special gas mixtures are used; most often, however, \(\varepsilon\) ranges from fractions of a percent to several percent. Efficiencies of the same order, and even somewhat smaller, are obtained for \(\gamma\)-rays[^14]. The geometrical factor \(G\) is somewhat less than 0.5 when measurements are carried out with solid samples placed under the window of an end-window Geiger–Müller counter. If the radioactive substance in the form of a gas is introduced into the detector, values of \(G\) close to unity can be achieved. The absorption factor \(f_a\) is almost always appreciably less than unity, and especially so in the registration of low-energy \(\beta\)-particles. In such cases it is necessary to take all precautions to reduce the influence of absorption, such as, for example, the use of very thin samples, the use of counters with thin windows or with no windows at all, etc. The contribution from the scattering effect differs from the contribution of the above-mentioned factors in that \(f_s\) can exceed unity, reaching in some cases values up to 1.75; this occurs, for example, when energetic \(\beta\)-particles from a sample mounted on a backing made of a material with a high atomic number are registered.

It is impossible here to dwell on the question of the exact determination of the counting factor even for the important case of \(\beta\)-particle registration. Fortunately, this subject has been the object of recent careful systematic investigations[^43], which have shown that \(A_t^\beta\) can be determined with a probable error of 2–4% for radioelements with a maximum \(\beta\)-particle energy lying between 0.3 and 1.0 MeV; at energies above 1 MeV the accuracy is 2% or better. In view of the large variation of \(\varepsilon\) depending on the character of the radiation (\(\beta\)-particles, \(\gamma\)- or X-rays), the greatest sensitivity in activation analysis is obtained when \(\beta\)-particles are registered, provided that no special apparatus is used for registration[^9]. The conclusion that follows from these necessarily brief remarks is thus that the choice of the counting instrument and of the measurement procedure, including the preparation and mounting of the sample, is determined by the radioelement used for the analysis.

Finally, in considering equation (5), it should be pointed out that the absolute mass of the impurity can be estimated in the case when, in addition to knowledge of the absolute decay rate \(A_t\), the beam intensity, the reaction cross section, and the half-life are also known. The determination of the last two quantities presents no special difficulties, and increasingly accurate values of these quantities are continually appearing in the literature[^33,^40]. Accurate measurement of the beam intensity, on the other hand, poses a number of difficult problems, especially when charged particles obtained from a cyclotron are used: protons, deuterons, \(\alpha\)-particles. The method for measuring the absolute value

density of thermal neutrons produced by a radium-beryllium source surrounded by a thick layer of paraffin has been described earlier².

If equation (5) is used for the absolute determination of mass, one more precaution must be borne in mind which must be observed. Equations (6) relate the measured number of pulses to the true or absolute number of radioactive decays only in the case where the β-spectrum is simple, i.e., if in each decay one β-particle or one γ- or X-ray quantum is emitted. Otherwise, it is necessary to use the total counting coefficient, defined as the ratio of the total number of registered pulses caused by the decay of the radioelement to the true number of decays. The total counting coefficient, therefore, takes into account all possible modes of decay.

Let us consider, for example, a hypothetical nucleus decaying in 25% of cases with the emission of a 1.5 MeV β-particle and in 75% of cases with the emission of a 0.6 MeV β-particle, after which a γ-quantum of energy 0.9 MeV is emitted. If, for the experimental setup used, we assume that the counting coefficient of the γ detector is 0.23 for 1.5 MeV β-particles, 0.11 for 0.6 MeV β-particles, and 0.02 for 0.9 MeV γ-quanta, then the total counting coefficient will be

\[ (0.25 \times 0.23) + (0.75 \times 0.11) + (0.75 \times 0.02) = 0.155. \]

Obviously, determination of the total counting coefficient depends on the decay scheme of the radioelement being measured. At present, nuclear spectroscopy is in such a state that, of 700 known radioisotopes, the decay schemes of no more than half have been firmly established. But this branch of nuclear science too is developing intensively, and the precise determination of the properties of new isotopes is progressing at great speed. The development of this field of nuclear physics is of exceptional importance for nuclear chemistry, since radioactive substances can be used for carrying out absolute physical and biological measurements only on the condition of complete knowledge of the decay schemes.

Fortunately, the problems under discussion do not constitute an obstacle to applying the activation method to the solution of problems in chemical and isotopic analysis. Good accuracy can be achieved in the analysis of samples of unknown composition by carrying out comparative measurements with samples of known composition. These standard samples, whose composition should basically coincide with the composition of the sample under investigation, are irradiated under identical conditions and, if possible, simultaneously with the sample being studied. Just as in the tracer-atom method, identical chemical operations are carried out on the known and unknown samples, and the samples prepared from them for radioactive measurements are, as far as possible, made identical with respect to thickness, dimensions, etc. The number of pulses from

standard must be measured in the same position relative to the counter as the investigated sample. Comparison of the counting rates obtained in this way, with correction for the measurement time (if the decay is appreciable), makes it possible to estimate the mass of the element under investigation by the formula

\[ \frac{\text{Total activity of element } X \text{ in the substance under investigation}} {\text{Total activity of element } X \text{ in the standard}} = \frac{\text{mass of } X \text{ in the substance under investigation}} {\text{mass of } X \text{ in the standard}} . \tag{7} \]

In the case where mass measurements are being made of samples of different composition, it may prove desirable to construct a calibration curve with the aid of a series of standards. As a rule, however, one properly chosen standard sample is quite sufficient for carrying out this analysis.

When the element being analyzed is present in quite negligible amounts, it may prove expedient to add a small amount of the natural element to the sample after its irradiation; the added substance serves as a “carrier” and facilitates the chemical separation of the desired radioisotope. Solid compounds containing radioisotopes are usually the most convenient; the basic requirements imposed on them are purity and a quite definite composition. The ratio of the mass of the measured sample to the initial mass of the added carrier determines the chemical yield of the separation and purification operation. If chemical losses occur, a correction may be introduced into the measured number of pulses from the investigated and standard samples with the aid of chemical-yield data. A difference in the total weight of the samples may also require the introduction of corrections for self-absorption of radioactive radiations. This correction, which can best be determined empirically, is especially important in the measurement of preparations emitting slow \(\beta\)-particles[^22].

If it is assumed that the various correction factors can be determined with sufficient accuracy, and that the treatment of the standard and of the investigated preparations has been carried out in practically identical fashion, then the principal error in the comparative method will be determined by the error in determining the number of pulses. A discussion of the errors due to the statistical nature of radioactive decay may be found in work[^21]. The experience accumulated up to the present indicates that the determination of the amount of impurities in unknown compositions can be carried out without special difficulty with an error of less than \(10\%\).

The comparative method of activation analysis may be briefly described as follows: a preparation of the substance under investigation and a standard preparation containing a known mass of the element being analyzed are irradiated. The samples are then dissolved and added to

a known amount of the element whose radioactivity is to be determined, after which the added element is chemically separated from the rest of the sample mass and from possible radioactive impurities of other elements contained in the sample. In doing this it is necessary to determine the chemical yield of this procedure. The activities of the substance under investigation and of the standard substance are then compared. To check the radiochemical purity of the measured substances, it may be expedient to measure their half-lives or the penetrating power of their radiations.

THE PRESENT STATE OF THE METHOD OF ACTIVATION ANALYSIS

The significance of the method of radioactivation for analytical chemistry lies in its exceptional accuracy and general applicability in solving problems of microanalysis. The further introduction of this method of analysis will undoubtedly depend on how accessible the means of nuclear physics used for the production and measurement of radioactive substances prove to be. The situation with apparatus for measuring radioactivity should be regarded as quite satisfactory. At present, installations are commercially available that make it possible to detect and measure quite effectively almost any of the three kinds of radiation emitted in radioactive decay. Further improvements in this field are still needed so that, in the future, radioactive measurements may become a purely technical operation. Such a tendency, apparently, is in fact being observed. On the other hand, the production of apparatus for the laboratory preparation of radioactivity on a scale sufficient for analytical and general-chemical purposes is still only in an embryonic state. It would perhaps be possible to stimulate the technical development of this question if a definite solution could be reached as to the type of nuclear projectile (slow neutrons or energetic charged particles) that makes it possible to obtain radioactivity in laboratory conditions with the greatest advantage. Here we can discuss this question from the point of view of activation analysis only in a few words.

The obvious criteria for the advisability of applying the method of activation analysis are its sensitivity and convenience. Thus, for example, the method of radioactivation will prove inconvenient in those cases when, upon irradiation of the element sought, radioelements with a long half-life are obtained, since in that case, as is evident from equation (5), an excessively long bombardment time would be required. However, if an intense beam of nuclear particles can be obtained, or if the cross section of the nuclear reaction proves to be sufficiently large, good sensitivity can be achieved even with short bombardment.

...processing. For rough orientation it may be considered desirable that the radioactivity generated have a half-life between 3 minutes and 100 hours. Owing to the circumstance that, on the average, there are about three radioactive isotopes for each stable isotope of the periodic table, serious difficulties in the choice of a suitable radioactive indicator are unlikely to arise. Thus, the practicability of the method, insofar as sensitivity is concerned, will be determined chiefly by the intensity of the beam and by the cross section of the reaction. These latter quantities are not quite independent, since the magnitude of the activation cross section \(\sigma_{\text{ak}}\), necessary for obtaining a given sensitivity, determines the required beam intensity.

On the other hand, the nature and intensity of the required beam determine the type and dimensions of the nuclear apparatus. It is therefore of interest to consider the dependence of \(\sigma_{\text{ak}}\) on the nature and energy of the nuclear projectile. For a better acquaintance with these details it is expedient to recall the definition of the concept of the activation cross section \(\sigma_{\text{ak}}\).

Let us consider a thin layer of a substance containing \(N\) nuclei per square centimeter. Let a flux of \(f\) elementary particles fall on one square centimeter of this layer. If, as a result of some reaction, \(N'\) atoms of the target undergo a certain nuclear reaction with the formation of a new radioelement, then the activation cross section for the given reaction is defined by the equation

\[ N' = f\sigma_{\text{ak}}N. \]

This definition assumes that the target is so thin that \(f\) practically does not change with the depth of penetration.

Since, in a first approximation, nuclear reactions may be regarded as a process of collision, it may be expected that the cross section is proportional to the geometrical area of the target atom. Thus one should expect that the cross sections will be of the order of \(10^{-24}\ \text{cm}^2\) per atom, and precisely such values are in fact encountered. The bombarded nucleus, of course, possesses a positive charge \(+Ze\), acting as a barrier to effective collisions with the bombarding particles. For this reason protons, deuterons, or \(\alpha\)-particles have to be accelerated to energies sufficient for their penetration into the nucleus. Although the dependence of the yield of a nuclear reaction on the energy of charged particles can be predicted only on the basis of detailed knowledge of the binding energy of the nucleus, the character of the potential barrier, the probability of competing reactions, and other factors discussed in detail by other authors \(^{4,5,23}\), we shall nevertheless note here certain very general factors that are important for obtaining artificial radioactivity. For reactions with charged

there exists, for particles, a rather sharp energy threshold below which the cross section is either very small or equal to zero. This threshold increases as the atomic number of the target atom increases. Above the threshold the reaction yield grows almost exponentially with energy, following the probability of the particle penetrating the nuclear potential barrier. At still higher energies the cross section begins to tend toward the value of the geometrical cross section of the nucleus and then either remains constant or begins to decrease if new, competing nuclear reactions arise. A fairly typical curve of the dependence of the cross section on energy, known as the excitation function, is shown in Fig. 1; this curve refers to the reaction of formation of the well-known radioelement \(\mathrm{Na}^{24}\) (14.8 hours) under bombardment of sodium by deuterons \({}^{12}\). At a given energy the reaction yield with deuterons is usually higher than with protons or \(\alpha\)-particles. This superiority of the deuteron as an agent for exciting nuclear reactions is partly explained by the fact that the reaction products are often neutrons. Events in which charged particles are reaction products are usually less probable than reactions with neutron emission, since charged particles must overcome the potential barrier of the residual nucleus. From this point of view, the interaction of neutrons with target nuclei should not depend on their energy, provided that charged particles are not reaction products. The cross section should be close to the geometrical dimensions of the target nucleus, which is in fact observed, with the exception of neutrons of very low energy, for which the interaction is especially strong. Neutrons having velocities corresponding to room temperature, i.e. approximately \(0.025\) eV, behave in accordance with the laws of quantum mechanics as particles of considerably larger dimensions compared with high-energy neutrons. To estimate the cross section one may use the de Broglie relation between wavelength and momentum:

Fig. 1. Excitation curve of the reaction of formation of \(\mathrm{Na}^{24}\) (14.8 hours) under bombardment by deuterons accelerated in a cyclotron.

\[ \sigma \lessgtr \pi \lambda^{2}=\pi\left(\frac{h}{mv}\right)^{2}=\frac{\pi h^{2}}{2mE}=\frac{6.46\cdot 10^{-19}}{E}. \tag{8} \]

A simple calculation in this way leads to a value of the order of \(10^{-18}\ \text{cm}^2\) per atom for the capture cross section of a thermal neutron. In experiment, however, the capture cross sections of thermal neutrons turn out to be somewhat smaller and vary in an indefinite manner in passing from element to element and even from isotope to isotope of the same element, ranging from \(0.001\cdot 10^{-21}\) to \(100000\cdot 10^{-24}\ \text{cm}^2\) per atom.

Table I gives a summary of known cross sections (expressed in square centimeters per atom of the normal element) for thermal neutrons.

Fig. 2. Dependence of the total neutron cross section on neutron energy for natural iridium.
(The scale is double logarithmic. The resonance peaks stand out sharply.)

for thermal neutrons. Cross sections expressed in this manner give a more correct representation of the activation capacity for each given element than the so-called isotope cross sections, usually used in purely physical questions, since in the latter a correction for isotopic composition is automatically included. The atomic cross section is therefore equal to the product of the isotopic cross section by the relative abundance of the isotopes. As is generally the case for most nuclear constants, new, more accurate data on isotope cross sections and on the percentage composition of isotopes are continually appearing. The capture cross sections of slow neutrons are known for many elements with an accuracy up to \(\pm 10\%\).

The curves of the dependence of neutron cross sections on energy possess a number of properties not inherent in the corresponding cri-

for fast charged particles. In the thermal region of velocities the cross sections of neutrons increase inversely proportionally to the square root of the energy. At somewhat higher energies \((>1.0\ \text{eV})\) resonance peaks are often observed, in which the cross sections take on very large values, decreasing to their previous, relatively small value as the energy is further changed by several electron-volts. At energies exceeding \(1000\ \text{eV}\) (fast neutrons), the cross sections take on an approximately constant value of the order of the geometrical cross section. A typical curve illustrating the foregoing is given in Fig. 2.

In conclusion we shall note two generalizations following from the data of Table 1, in view of their importance for the application of thermal neutrons in activation analysis. First, analysis by activation with neutrons is impossible for light elements \((Z<10)\), owing to the exceptionally small absorption cross sections, and also because only short-lived radioelements are formed. Second, elements with odd atomic numbers are detected with much greater sensitivity than elements with even \(Z\), both because of their larger cross sections and because these elements are either monoisotopic, or, in the extreme case, contain no more than two stable isotopes.

The elementary discussion of nuclear cross sections set forth above makes it possible to estimate the order of magnitude of the intensity of the flux of nuclear projectiles required for measuring micrograms of one or another element by the method of activation analysis.

To make this estimate by means of equation (5), we shall assume that an activity of 10 disintegrations per second can be detected without particular difficulty with the required accuracy. In addition, we shall assume that the lifetime of the corresponding radioelements is sufficiently short that the irradiation time necessary to attain “half-saturation” \((S=0.5)\) is not too long. Let us take the mass of the element sought to be equal to 100. Finally, let us assume that the atomic activation cross section is equal to \(5\cdot10^{-26}\ \text{cm}^2/\text{atom}\); this quantity may be regarded as characteristic of various elements throughout the periodic table. A numerical calculation then shows that, to detect one microgram of the hypothetical substance under consideration with an accuracy of up to \(\pm 5\%\), a flux of \(6.6\cdot10^{10}\) particles/\(\text{cm}^2\cdot\text{s}\) is required. Flux values between \(5\cdot10^{10}\) and \(5\cdot10^{12}\) particles per \(\text{cm}^2\cdot\text{s}\) reflect those limits which are determined, respectively, by the most and least favorable combinations of cross sections and decay periods encountered in practice.

It is thus obvious that the practical feasibility of the method of activation analysis is determined by the possibility of obtaining

Table I

Summary of atomic cross sections for activation upon capture of thermal neutrons

(values of $\sigma_{\text{ak}}$ are given in $10^{-24}\ \text{cm}^2$ per atom of the element and refer to neutrons with energy $0.025\ \text{eV}$)*

* The data have been calculated on the basis of values given in $^{40}$.

Continuation of Table 1

Table 1 (continued)

flows intensities from \(10^{10}\) to \(10^{12}\) particles/\(\mathrm{cm}^2\cdot\mathrm{sec}\), and therefore the remainder of this paragraph will be devoted to considering the present state and further possibilities of obtaining such fluxes.

The cyclotron is undoubtedly the best-known and, perhaps, from the standpoint of activation analysis, the most important source for obtaining energetic charged particles. To give an idea of the fluxes of fast nuclear particles that can be obtained with this machine, let us note that a beam of protons or deuterons with a current of \(1\ \mu\mathrm{a}\) corresponds to a flux of \(6.3\cdot 10^{12}\) ions per second, and in modern machines with constant frequency cases are not rare in which the current reaches the order of several hundred microamperes. In a typical case, under irradiation with a beam of current \(100\ \mu\mathrm{a}\) and cross section \(10\ \mathrm{cm}^2\), nuclei distributed over such a surface are subjected to the action of a comparatively intense flux of approximately \(6\cdot 10^{13}\) particles/\(\mathrm{cm}^2\cdot\mathrm{sec}\). The slowing down of such a quantity of charged particles with energies of \(5\)—\(10\ \mathrm{Mev}\) in several millimeters of target leads to the liberation of several kilowatts of energy, which must be removed in a corresponding manner in order to avoid serious losses of radioactivity as a result of evaporation.

Special attention must be paid to this circumstance if nonmetallic samples are irradiated; some assistance may be provided by the data of recent investigations\(^{27}\).

In activation with the aid of a cyclotron, exceptionally high sensitivities may be expected. For illustration let us make a rough estimate of the minimum concentration of sodium impurity that can still be detected in a sheet of aluminum of cross section \(1\ \mathrm{cm}^2\). We shall assume that the following conditions obtain, which are apparently sufficiently characteristic for such cases.

The preparation is subjected to the action of a beam of \(5\ \mathrm{Mev}\) deuterons of intensity \(100\ \mu\mathrm{a}\) for 12 minutes, after which it is allowed to stand for 30 minutes so that the strong 2.3-minute activity formed in aluminum by the reaction \(\mathrm{Al}^{27}(d,p)\mathrm{Al}^{28}\) has practically disappeared. Let us suppose that sodium is the only impurity in the aluminum, and therefore no chemical separation will be required. Then the aluminum target can be placed directly under a counter (or any other suitable detector) and counts taken. Let us assume that the number of pulses proved to be 100 per minute, or 1.67 per second, and that this activity is ascribed to the isotope \(\mathrm{Na}^{24}\) with a period of 14.8 hours, produced as a result of the reaction \(\mathrm{Na}^{23}(d,p)\mathrm{Na}^{24}\). With correction for decay, which continued for 45 minutes from the moment the deuteron bombardment ended, the number of pulses per second is now, in accordance with the equation

(4) will be equal to 1.73. To calculate the decay rate \(A_t\), it is necessary to know the total counting factor. Fortunately, the decay scheme of \(Na^{24}\) has been established quite satisfactorily\({}^{39}\). If we assume that the geometric factor in measuring the number of pulses from the aluminum foil was \(30\%\) and that for \(\beta\)-particles with energy \(1.39\) MeV \(\varepsilon = 1.0\), \(f_a = f_s = 1.0\), while for the simultaneously emitted \(\gamma\)-quanta with energies \(1.39\) MeV and \(2.76\) MeV \(\varepsilon = 0.012\) and \(\varepsilon = 0.023\), with \(f_a = f_s = 1.0\), then for the total counting factor we obtain:
\[ (1.0 \times 1.0 \times 0.3 \times 1.0 \times 1.0) + (1.0 \times 0.012 \times 0.3 \times 1.0 \times 1.0) + (1.0 \times 0.023 \times 0.3 \times 1.0 \times 1.0) = 0.31. \]
It follows that
\[ A_t = \frac{1.73}{0.31} = 5.58 \]
decays per second. Substituting this and the following values into equation (5): \(M = 23\); \(\theta = 1.00\), \(f = 6 \cdot 10^{13}\), \(\sigma_{\mathrm{ak}} = 4.5 \cdot 10^{-25}\) (from Fig. 1), \(S = 9.3 \cdot 10^{-3}\), we obtain:
\[ \text{mass of sodium in grams} = \]
\[ = \frac{5.58 \cdot 23}{6.02 \cdot 10^{23} \cdot 6 \cdot 10^{13} \cdot 4.5 \cdot 10^{-25} \cdot 9.3 \cdot 10^{-3} \cdot 1.00} = 0.5 \cdot 10^{-10}. \]

With an aluminum-foil thickness of \(0.025\) cm, the weight of unit area is \(6.8\ \mathrm{mg}\); hence the sodium concentration proves to be of the order of \(0.1 \cdot 10^{-6}\).

In view of a number of uncertainties, the estimate made above should be regarded as correct only as to order of magnitude. In the example of activation analysis considered, the target must be thin so that \(f\) is constant. Probably, under real conditions of using deuteron beams for analytical purposes, the aluminum thickness will be such that the deuterons will be completely absorbed. Under these conditions a lower specific activity will be obtained, although the total activity will be at least an order of magnitude higher owing to the more efficient use of the beam. Thus, the sensitivity of the method worsens when thick targets are used, especially if chemical separation operations are also required. In practice, a comparative method will probably be applied, in which known and unknown samples are irradiated simultaneously. Nevertheless, calculations such as those given above may prove useful for establishing the limits of sensitivity possible when using beams of particles of known maximum energy from a cyclotron.

Another method of irradiation by means of a cyclotron consists in the formation of fast secondary neutrons in the target by bombarding it with primary charged particles. These fast neutrons can then be used to produce the desired nuclear reaction, or they can first be slowed down to thermal energies with the aid of some hydrogen-containing substance and then used. As targets for

for obtaining neutrons in a cyclotron light elements are used, since even under bombardment by protons and deuterons of comparatively low energy the yield is large. Two nuclear reactions giving an appreciable neutron yield are:

\[ {}_{4}\mathrm{Be}^{9}+{}_{1}\mathrm{H}^{2}\to{}_{5}\mathrm{B}^{10}+{}_{0}n^{1}+3.7\ \text{Mev}, \tag{9} \]

\[ {}_{3}\mathrm{Li}^{7}+{}_{1}\mathrm{H}^{2}\to{}_{4}\mathrm{Be}^{8}+{}_{0}n^{1}+15.2\ \text{Mev}. \tag{10} \]

The second reaction is used in those cases when neutrons of very high velocities are required. To obtain a source of thermal neutrons with the aid of a cyclotron, beryllium targets are usually preferred, owing to their low volatility and good thermal conductivity. As follows from a recent communication[^36], by bombarding beryllium with deuterons of energy \(10\) Mev one can obtain very intense sources, up to \((3.7\pm0.4)\cdot10^{10}\) neutrons per second for each microampere. This neutron flux, however, is considerably less intense than the beam of charged particles in the cyclotron. Since in the latter \(1\ \mu\text{a}\) corresponds to \(6.3\cdot10^{12}\) particles per second, consequently, for each fast neutron formed there are 170 primary energetic deuterons. One more no less important factor must be taken into account in making such comparisons. Namely, in the cyclotron a directed beam of charged particles is obtained, and the transverse cross section of the beam can be made small; these circumstances substantially improve the geometrical conditions for producing artificial radioactivity. Practically all fast charged particles in the cyclotron strike a region of the target only a few centimeters in area. A beryllium target, on the other hand, emits neutrons isotropically when bombarded by deuterons. It is evident that in this case only a small fraction of the total number of fast neutrons falls on a unit area of the target.

Suppose, for example, that when a beam of deuterons with energy \(10\) Mev and current strength \(100\ \mu\text{a}\) falls on the face of a cube of metallic beryllium \(2\) cm thick, a point source of neutrons is obtained; then \(3.7\cdot10^{12}\) neutrons per second are emitted into the solid angle \(360^\circ\). A disk of area \(1\ \text{cm}^{2}\), placed immediately behind the rear wall, will intercept approximately \(2\%\) of all fast neutrons, i.e. it will be subjected to a neutron flux of intensity approximately \(7.4\cdot10^{11}\) neutrons/\(\text{cm}^{2}\cdot\text{sec}\). Thus, in order to obtain radioactivities comparable with those obtained with the same beam of deuterons acting directly on the target, it is necessary that the neutron cross sections be correspondingly greater by \(10^{3}\)—\(10^{4}\) times.

In view of the fact that the capture cross sections of slow neutrons exceed the cross sections for energetic charged particles, consideration of the order of magnitude of thermal-neutron fluxes obtained in the cyclotron has a direct bearing on the question of the prospects of the activation-analysis method. Usually such sources of slow neutrons are obtained by slowing down fast neutrons with the aid of thick layers of paraffin, placed as close as possible to the beryllium target and surrounding it on three sides. The slowing-down process requires on the average 20–30 collisions of the given neutron with the hydrogen atoms of the paraffin. Consequently, already at a distance of approximately 3 cm from the fast-neutron source a quite sufficient slowing down is obtained. Subsequently the thermal neutrons diffuse through the paraffin until they are captured by impurities or by hydrogen (parasitic losses), or escape through the boundary (leakage losses). To prevent leakage of neutrons into the room, a thin layer of cadmium is placed along the boundaries. For protection against energetic γ-rays emerging from the target, a lead shield several centimeters thick is also placed at the periphery of the paraffin.

One may approximately estimate the intensity of the flux of slow neutrons passing through a thin foil of unit area, placed at a distance of 5 cm from a source in a cyclotron emitting \(3.7 \cdot 10^{12}\) neutrons per second; for this it is sufficient to calculate the geometrical factor for this distance and to assume that the flux of slow neutrons is, roughly, equal to the flux of fast neutrons in the absence of paraffin. Thus, for example, through an aluminum foil located at the indicated distance from the source there would pass a flux of approximately \(10^{10}\) neutrons per \(\text{cm}^2\) per sec. In principle, analysis for the presence of sodium impurities can be carried out with the aid of the neutron-capture reaction \(\mathrm{Na}^{23}(n,\gamma)\mathrm{Na}^{24}\), which gives \(\mathrm{Na}^{24}\) with its characteristic period of 14.8 hours. In this case the sensitivity would be considerably better than in the case of direct irradiation of the sample by deuterons at 10 MeV, provided that in all other respects the methods of analysis are identical. The point is that although the activation cross section for the capture of slow neutrons by sodium, equal to \(4.5 \cdot 10^{-25}\ \text{cm}^2/\text{atom}\), approximately coincides with the corresponding value for fast deuterons, nevertheless, to obtain equal activity with deuterons a flux 6300 times more intense would be required. Thus, if the element being analyzed does not possess a neutron-capture cross section exceeding by at least a thousandfold the cross sections of reactions with fast charged particles, then direct bombardment in the cyclotron should be preferred. In heavy elements neutron reactions are more advantageous owing to the presence of a high barrier for

charged particles. Moreover, in a number of specific cases reactions with slow neutrons may prove more advantageous also for the reason that a negligible amount of heat is evolved and recoil effects are almost always absent.

For the activation of light elements one can use low-energy deuterons, which makes it possible to employ simpler and more accessible apparatus, such as, for example, electrostatic generators.

As a useful illustration of the use of a low-voltage nuclear accelerator for the rapid and accurate analysis of small amounts of carbon in iron, one may cite the following recently published work.^3 The samples for analysis were irradiated for 10 minutes with a beam of \(0.8\) MeV deuterons or protons with a current of \(20\ \mu\text{A}\). The following reactions were obtained:

\[ {}_{6}\mathrm{C}^{12}+{}_{1}\mathrm{H}^{2}\to{}_{7}\mathrm{N}^{13}+{}_{0}n^{1}, \tag{11} \]

\[ {}_{6}\mathrm{C}^{12}+{}_{1}\mathrm{H}^{1}\to{}_{7}\mathrm{N}^{13}+\gamma. \tag{12} \]

In order to reduce to a minimum the inevitable fluctuations in the operation of the accelerator, the sample with an unknown amount of carbon and a standard of silicon carbide (\(\sim 30\%\) carbon) were attached to a circular disk which, while rotating, periodically crossed the particle beam; as a result, both samples were alternately irradiated in the same way. The measured activities of 9.93-minute \(N^{13}\) in the unknown and standard samples were compared; the carbon content in the first sample was then calculated with the aid of equation (7). Table II gives a comparison of the results obtained by the nuclear-chemical method with the results of ordinary chemical analysis of identical samples. From this comparison it can apparently be concluded that the present accuracy of the activation method permits the analysis of metallic samples containing carbon in concentrations of the order of \(0.05\%\). In principle, the activation-analysis method should be more sensitive by at least one order of magnitude; this can be verified by a simple calculation on the basis of equation (5).

Suppose, for example, that the positron activity of \(N^{13}\) was determined 5 minutes after bombardment and that the minimum acceptable number of pulses is 1000 per minute, or 16.7 per second. Correction for the decay that occurred during the time before measurement gives 23.7 pulses per second for a half-life of 9.93 minutes. On the basis of the known decay scheme^34 and with a geometrical factor for a Geiger–Müller counter of 30%, we obtain the following estimate of the total counting factor:
\[ (1.0\times 1.0\times 0.3\times 1.0\times 1.0)+(2.0\times 0.032\times 0.3\times 1.0\times 1.0)=0.32. \]
Referring to the previously published^25 excitation curve for the reaction \(\mathrm{C}^{12}(d,n)\mathrm{N}^{13}\), one can see that at an energy of \(700\) keV

the average cross section is of the order of \(10^{-27}\ \text{cm}^2/\text{atom}\). Consequently, the mass of carbon in grams is equal to:

\[ \frac{12\cdot 23.7/0.32}{6.02\cdot 10^{23}\cdot 1.2\cdot 10^{13}\cdot 10^{-27}\cdot 0.5\cdot 1.0} = 2\cdot 10^{-7}. \]

An idea of the enormous sensitivity inherent in this method of carbon analysis can be formed from the special precautions that were taken in the cited work to degrease the metal surfaces and to preserve their cleanliness up to the moment of bombardment. Since the penetration depth in iron of deuterons or protons of relatively low energy is only of the order of \(1.5—2.0\) microns, even a monomolecular layer of fats would be sufficient to create the appearance of a carbon content of the order of several hundredths of a percent. And indeed, in this work the most serious danger of carbon contamination was due precisely to the vapors of the grease in the vacuum chamber. An attempt to circumvent this difficulty by covering the samples with thin aluminum foil was crowned with only partial success, since ions of carbon molecules formed in the deuteron beam passed through these coatings. As Seaborg indicated \(^{31}\), who investigated the technique of working with the cyclotron, recoil effects and losses due to evaporation are the cause of annoying, and sometimes, at high energies, very considerable difficulties.

In order that emphasizing the necessity of using intense fluxes not create the impression that this factor is the only one determining the sensitivity in activation analysis, it should be noted that some high-energy nuclear machines, such as, for example, the betatron and the synchrotron, which are capable of creating large fluxes of fast electrons or penetrating \(\gamma\)-rays, are significantly less effective than the cyclotron or modern sources of thermal neutrons with respect to the creation of radioactivity. This circumstance is explained by the fact that the probability of nuclear reactions due to capture by the nucleus of electrons or photons is many thousands of times less than the probability of a reaction caused by protons, deuterons, \(\alpha\)-particles, or neutrons.

The appearance of a low-voltage neutron generator is an important step toward the manufacture of nuclear machines of medium cost for producing radioactivity. The nuclear reaction used in this case is the following:

\[ {}_{1}\mathrm{H}^{2}+{}_{1}\mathrm{H}^{2}\to {}_{2}\mathrm{He}^{3}+{}_{0}n^{1}+2.4\ \text{MeV}. \tag{13} \]

Here a target of heavy ice (or of paraffin containing deuterium) is bombarded with deuterons of energy \(200\ \text{keV}\). A very abundant emission of neutrons is obtained at deuteron-beam currents of the order of several hundred microamperes. Such a

Table II

Determination of the carbon content in iron by means of activation analysis

Irradiation of a specimen of dimensions $30 \times 30 \times 0.2$ mm and of a silicon carbide standard containing 30% carbon. Bombardment with particles of energy 800 keV at a current of 20 μA. Exposure: one minute with deuterons and 10 minutes with protons.

ACTIVATION ANALYSIS

The installation38 produced beams of up to \(8 \cdot 10^7\) neutrons per second, which is approximately equivalent to the number of neutrons obtained from \(10\) g of a \((\mathrm{Ra}\text{-}\mathrm{Be})\) preparation. In general, installations of this type will be important for obtaining radioactive labeled atoms. If it proves possible to increase the intensity of the beams by an order of magnitude, then such installations will undoubtedly become an important tool for activation analysis as well.

For completeness we shall also say a few words about the above-mentioned \((\mathrm{Ra}\text{-}\mathrm{Be})\) neutron source. Despite its low intensities, this source may prove useful in individual cases, such as, for example, in the activation analysis of certain rare earths contained in natural minerals, or for detecting noble metals in platinum ores.

The fast neutrons of this source are produced as a result of the bombardment of beryllium by energetic \(\alpha\)-particles arising in the decay of radium and its daughter radioactive elements:

\[ {}_4\mathrm{Be}^9 + {}_2\mathrm{He}^4 \to {}_6\mathrm{C}^{12} + {}_0 n^1 + \text{from } 1.0 \text{ to } 6.0 \text{ MeV}. \tag{14} \]

The energy distribution of the fast neutrons is very complex, since the compound nucleus \(\mathrm{C}^{13}\) has several different excited levels corresponding to different interaction energies; moreover, the spectrum of \(\alpha\)-particles from radium and from the radioelements in equilibrium with it is also complex in character. The neutron yield is proportional to the mass of radium for a given quantity of beryllium; if, however, the quantity of radium is fixed, then with increasing quantity of beryllium the yield increases in such a way that it proves advantageous to admix the largest possible portions of the latter. Because of the increase in volume, one must choose a certain compromise ratio between the masses of beryllium and radium. Most often this ratio is chosen between 3 and 10.

The neutron yield in a pressed \((\mathrm{Ra}\text{-}\mathrm{Be})\) mixture of known composition can be estimated from the formula1:

\[ \text{number of neutrons per second} \simeq 1.3 \cdot 10^7 \frac{M_{\mathrm{Ra}} \cdot M_{\mathrm{Be}}}{M_{\mathrm{Ra}} + M_{\mathrm{Be}}}; \tag{15} \]

where \(M_{\mathrm{Ra}}\) and \(M_{\mathrm{Be}}\) are the masses of radium and beryllium, respectively, in grams. Recent measurements of the intensity of a source consisting of \(500\) mg of radium (in the form of radium bromide) and \(300\) mg of beryllium (in the form of metallic powder) gave the value \((5.5 \pm 0.4)\cdot 10^6\) neutrons/sec, which agrees well with equation (15).

The emission of a considerable number of high-energy \(\gamma\)-quanta, as well as the emission of \(\alpha\)-particles by radium, creates significant difficulties in working with neutron sources of the type under consideration. Therefore, in those cases where the presence of \(\gamma\)-rays may prove undesirable, one uses the natural radioactive isotope of polonium, which emits \(\alpha\)-rays and only a small quantity of

γ-quanta. Unfortunately, the short half-life of Po\(^{210}\) (140 days) leads to a noticeable decrease in the neutron yield with time.

In prospect there is the possibility of preparing neutron sources from artificially radioactive elements. The most promising, perhaps, is the photodisintegration reaction of Be\(^9\) by γ-quanta with energy exceeding 1.63 MeV:

\[ {}_{4}\mathrm{Be}^{9}+h\nu \to {}_{4}\mathrm{Be}^{8}+{}_{0}n^{1}. \tag{16} \]

Thus, in this case radioactive elements must satisfy the requirement of a sufficiently intense emission of γ-rays with energy greater than 1.6 MeV. It is also desirable that their half-life be long. Further, in order to obtain intense preparations, as, for example, when irradiating in a nuclear pile, it is also necessary that the neutron-capture cross sections be sufficiently large. Measurements of the yields of photoneutron sources consisting of radioelements obtained in a pile \({}^{28}\) indicate that 60-day Sb\(^{124}\) may prove to be a suitable element for these purposes. The observed yield, \(3.2\cdot 10^{6}\) neutrons/sec. per 1 curie, is not much worse than the yield of the ordinary (Ra-Be) source. The volume of an antimony-beryllium source is also found to be satisfactory in comparison with a (Ra-Be) source. The principal drawback is due to the two-month period. The neutrons from an (Sb-Be) source are monochromatic and have an energy of 24 keV.

The simplicity of the preparation and use of radium-beryllium and antimony-beryllium neutron sources is so considerable that the possibility of their broad application in radioactivity studies undoubtedly deserves closer attention from chemists. In some cases, when ordinary analytical methods are accompanied by considerable difficulties of chemical separation, as occurs, for example, in compositions containing rare earths, direct analysis by activation with a “natural” neutron source may prove more advantageous. As an example, we note that it has been possible to determine the quantity of dysprosium in yttrium at a concentration below 0.01% by activation with a (Ra-Be) source weighing 600 mg. In a number of metallurgical analyses, where it is seldom necessary to determine concentrations of alloy components below \(10^{-3}\%\), one may use an (Sb-Be) neutron source placed at the center of a paraffin moderator having a simple geometry. Calculation shows that it is possible to detect less than 0.1% rhodium and 0.01% iridium in alloys with platinum.

In conclusion, mention should be made of the possibility of using natural or photoneutron sources for detecting elements that strongly absorb thermal neutrons but, for one reason or another, do not become radioactive. In this variant of the activation-analysis method, the unknown preparation is placed between a source of slow neutrons and a thin

foil made of indium or rhodium. The activity of the indium is then compared with the activity of an analogous foil obtained by screening the standard in the same position in which the unknown preparation was located. The use of this method for determining 0.1% boron in silica glass with an absolute accuracy of 0.015% was described recently by Martell and Su^24.

ACTIVATION ANALYSIS BY MEANS OF A NUCLEAR REACTOR

Owing to the possibility of obtaining intense fluxes of thermal neutrons over a large spatial extent, the nuclear reactor as an instrument of mass radioactivation has no rivals. In particular, the reactor surpasses the cyclotron as a source of neutrons not only with respect to intensity, but also with respect to the total magnitude of the effective flux. To illustrate this, one may cite certain data relating to the graphite and natural-uranium reactor operating at the National Laboratory at Oak Ridge. The maximum flux of thermal neutrons lies between \(1 \cdot 10^{12}\) and \(2 \cdot 10^{12}\) neutrons/\(\text{cm}^2 \cdot \text{sec}\). Experimental samples to be irradiated are placed at various points of the reactor. For short irradiations of quantities weighing less than one gram, pneumatic tubes are often used. The flux intensity in this method can reach, at the working point, up to 57% of the maximum value. Larger samples, subject to prolonged irradiation for days or even months, are placed in closed aluminum cans, which are inserted into special rectangular openings drilled perpendicular to the axes of the long graphite rods. Depending on the position in the reactor, the flux acting on such a sample varies between 25 and 30% of the maximum value. The geometrical factor is equal to 100%, since the target is located inside the reactor.

As can be verified by direct calculation from formula (5) and on the basis of the activation cross sections given in Table I, for elements that are not too light one can obtain very high sensitivities when irradiating with thermal-neutron fluxes of intensity between \(2.5 \cdot 10^{11}\) and \(5.7 \cdot 10^{11}\) neutrons/\(\text{cm}^2 \cdot \text{sec}\). We shall give here several examples illustrating the use of the reactor, since it is not excluded that this method may prove expedient in the activation analysis of certain special materials. It is necessary here to assume that the half-life period does not impose strict limitations.

The most primitive method of analysis by means of a reactor consists in irradiating an unknown sample for several minutes and determining the regularities of decay of the mixture of activities formed in this way. Then the half-life periods are determined

component by decomposing a complex decay curve into its elementary constituents. From the periods obtained and from tables of nuclear data, a qualitative analysis of an unknown sample can be performed. As an example of the application of this method, Fig. 3 shows the decay curve of a sample of pure rubidium carbonate irradiated in a reactor. Unmistakable indications of the presence of small amounts

Fig. 3. Qualitative analysis by the activation method. Decay curve of rubidium carbonate irradiated with neutrons in a reactor.

of potassium and cesium. The half-lives appearing in an unknown sample cannot always be identified. In this case Clark and Jones’s table \(^{10}\), which gives a list of elements in order of increasing decay periods, can be of great assistance.

A number of factors limit the applicability of this method. First, it may prove to require much time, since in some cases, in order to determine accurately constituents with a longest period, one has to carry out prolonged measurements of the total intensity. Further, as a rule, decomposition of a decay curve consisting of four or more components gives unreliable results because of the accumulation of large errors arising in

as a result of the preceding subtractions. In some cases, as, for example, in searches for small amounts of molybdenum in rhenium, yttrium in dysprosium, or copper in gold, the activity of the principal element may prove so large that without chemical separation it will be impossible to determine the decay law of the impurity. Finally, if the half-lives of two radioelements differ from one another by less than 50%, then it proves very difficult

Fig. 4. Absorption curves of the radiation of \(K^{42}\) (12.8 hours) and \(Na^{24}\) (14.8 hours) in aluminum. Experimental basis for applying the method of differential absorption to the analysis of mixtures of sodium and potassium carbonates.

to decompose a complex curve into its components with sufficient accuracy.

In some cases it is possible to perform quantitative activation analysis of binary alloys by using the difference in the radiation of the radioelements formed. As an example we shall cite the case of analysis of a mixture of sodium and potassium carbonates.

In the case under consideration, the half-lives of \(Na^{24}\) and \(K^{42}\), formed upon neutron bombardment, were respectively 14.8 hours and 12.4 hours. These values differ too little from one another for it to be possible to decompose the total decay curve into its components. However, the maximum energy of the \(\beta\)-particles of active sodium is 1.39 MeV, while for the \(\beta\)-particles of potassium the maximum energy is 3.58 MeV. As is evident from Fig. 4, the less energetic

...the β-radiation of sodium (Naβ) is completely absorbed by an aluminum layer of thickness 700 mg/cm², whereas the β-radiation of potassium (Kβ) disappears only when a layer of 1700 mg/cm² is placed between the specimen and the Geiger–Müller counter. The penetrating γ-rays of sodium and potassium are absorbed only very slightly by such thicknesses of aluminum. Let \(A_{700}\) and \(A_{1700}\) denote the total number of pulses measured with absorbers of thickness 700 and 1700 mg/cm², respectively, in the corresponding position; let \(K^\beta\), \(K^\gamma\), and \(Na^\gamma\) denote, respectively, the contribution of the β- and γ-radiation of potassium and of the γ-radiation of sodium. Then the following relations hold:

\[ A_{700}=K^\beta+K^\gamma+Na^\gamma, \tag{17} \]

\[ A_{1700}=K^\gamma+Na^\gamma. \tag{18} \]

After introducing a correction for the somewhat greater absorption of the γ-rays of \(Na^{24}\) and \(K^{42}\) when the thickness of the aluminum absorber is increased from 700 to 1700 mg/cm², and after subtracting equation (18) from equation (17), \(K^\beta\) can be found. Next, with the aid of a pure preparation of 12.4-hour \(K^{42}\), the ratio \(R\) of the number of pulses \(K^\gamma\) to \(K^\beta\) at a thickness of 700 mg/cm² is determined. Then

\[ K^\beta=A_{700}-1.024A_{1700}, \tag{19} \]

\[ K^\gamma=RK^\beta=R(A_{700}-1.02A_{1700}), \tag{20} \]

\[ Na^\gamma=A_{1700}-R(A_{700}-1.02A_{1700}) \tag{21} \]

or

\[ Na^\gamma:K^\gamma=\frac{A_{1700}}{R(A_{700}-1.024A_{1700})}-1. \tag{22} \]

To determine the ratio of the amounts of sodium and potassium in a mixture from the experimentally measured ratio of the number of pulses from γ-rays, corrected for decay from the moment at which bombardment ended, it is necessary to know the following: the number of γ-quanta emitted in each decay, the counting efficiency for γ-quanta \(\varepsilon\), and the ratio of the activation cross sections by slow neutrons.

For illustration, we give several results obtained in the course of a preliminary study of the method of differential absorption using a nuclear boiler.

Five preparations, each weighing 100 mg and consisting of a definite mixture of chemically pure carbonates of sodium and potassium, were sealed into separate quartz tubes 3 cm long and with an internal diameter of 4 mm. Each tube was then placed in a plastic box and sent to the end of the pneumatic tube, where irradiation was carried out for 5 minutes. Then each preparation was dissolved in distilled water, and in small glass pycnometers the volume was brought to 5 ml. With the aid of micropipettes, portions from 100 to 500 λ were taken; these preparations were then evaporated on watch glasses 2.5 cm in diameter.

To ensure a strictly defined geometry, the glass was attached to the middle of thin aluminum plates measuring \(6.35 \times 7.15\) cm. The value \(R\) was determined simultaneously with the measurement of the unknown samples and under the same conditions. It proved to be equal to 0.0277. The fraction of sodium in the mixture was determined from the equation

\[ \%\,\mathrm{Na}= \frac{3.75\cdot 10^{-2}(\mathrm{Na}^{\gamma}/\mathrm{K}^{\gamma})_0} {1+3.75\cdot 10^{-2}(\mathrm{Na}^{\gamma}/\mathrm{K}^{\gamma})_0}\cdot 100 . \tag{23} \]

A comparison of the compositions measured by this method with the values obtained on the basis of gravimetric measurements made during preparation of the mixtures is given in Table III.

Table III

Application of the differential-absorption method for the analysis of neutron-activated mixtures of sodium and potassium

The method described obviously has the advantages of speed and simplicity. However, a number of disadvantages are also evident. Apart from the fact that the ratio \(\mathrm{K}^{\gamma}/\mathrm{K}^{\beta}\) must be measured under conditions strictly identical with the conditions of measurement of the unknown preparation, consideration of equation (22) shows that, because of the comparatively small difference in the number of pulses under absorbers of 700 and 1700 mg/cm\(^2\), respectively, small errors in individual activity measurements may lead to considerable errors in the final result. Finally, the method under discussion will be inapplicable if other radioactive contaminants are present, especially if the latter emit fast \(\beta\)-particles. In this case chemical separation becomes necessary; however, then the principal advantage of the differential-absorption method over the comparison method in activation analysis is generally lost.

As a final example of the use of the pile for activation analysis, we shall describe an absolute method for determining the amount of manganese in 3S aluminum.

A carefully cleaned and weighed small plate of aluminum was first irradiated for 5 minutes, after which a complex decay curve was taken. This curve is shown in Fig. 5. Analysis of the curve indicated the presence of small quantities of sodium impurities, activated by the reaction \(\mathrm{Na}^{23}(n,\gamma)\), 14.8-hour \(\mathrm{Na}^{24}\); in addition, strong activity of \(\mathrm{Al}^{28}\) (2.4 min.) and \(\mathrm{Mn}^{56}\) (2.59 hours) was observed. Despite the fact that approximately 5% of the neutrons at the end of the pneumatic tube have energies extending from the threshold of the cadmium screen (0.18 eV) up to

Fig. 5. Analysis of a manganese–aluminum alloy by radioactivation. Decay curve of the \(\beta\)- and \(\gamma\)-activity of 3 g of aluminum irradiated with neutrons from the reactor.

several million electron-volts, no competing reactions of the type \(\mathrm{Al}^{27}(n,p)\mathrm{Mg}^{27}\) (10.2 min.) or \(\mathrm{Al}^{27}(n,\alpha)\mathrm{Na}^{24}\) (14.8 hours) were nevertheless observed. The extrapolated activity of 2.6-hour \(\mathrm{Mn}^{56}\) (Fig. 5) and the corresponding bombardment and recording data lead to the conclusion that the aluminum–manganese alloy contained about 1% manganese.

For a more accurate determination of the manganese content in the reactor, a second alloy sample was irradiated. In this experiment the flux meter was irradiated simultaneously with a pure sample weighing exactly 25 mg for 5 minutes. Calibration of the meter by means of a calibrated ionization chamber\({}^{17}\) made it possible to determine precisely the intensity of the thermal-neutron flux during irradiation. The value obtained was \(6.4 \cdot 10^{11}\) neutrons/\(\mathrm{cm}^{2}\cdot\)sec. In order to eliminate the strong activity in aluminum, all further operations were temporarily interrupted for

30 minutes. After this the alloy was dissolved, and the active manganese was separated with the aid of 1 mg of manganese carrier. This procedure is analogous to that described by Clark and Overman[^11]. The final precipitate, containing manganese, was dissolved in two drops of concentrated hydrochloric acid and, with a micropipette, was quantitatively transferred to a sample mount.

The latter was a polystyrene film of thickness 2.6 mg/cm\(^2\). After application to the film, the sample was slowly evaporated to dryness under an infrared lamp. Immediately before completion of the evaporation, a drop of aluminum hydroxide was added to convert the soluble manganese chloride into the oxide hydrate. The polystyrene film was then cemented to a standard aluminum plate with polystyrene cement; all precautions were taken to ensure that the precipitate was located exactly above the center of the hole in the plate, of diameter 2.5 cm. The sample was covered from above with a similar polystyrene film to protect the counting setup from accidental contamination.

The sample prepared by the method described was then measured with an end-window Geiger–Müller counter with a window thickness of 3 mg/cm\(^2\) and with strictly defined geometry (2.95%). Counting was carried out for a period sufficient to obtain a statistical error not exceeding 2%. The total number of pulses, corrected for efficiency, background, and the number of pulses from \(\gamma\)-rays, was 8420 pulses per minute, which, taking into account a chemical yield of 88%, gives 9570 pulses/min, or 159.5 pulses/sec. The total counting factor for \(\beta\)-decay was estimated on the basis of the known decay scheme[^13,^15] of Mn\({}^{56}\) and the previously estimated values of the absorption and scattering factors \(f_A\) and \(f_s\), as follows:

\[ Y_{\text{total}}=(0.60\times0.0295\times0.96\times1.04)+(0.25\times0.0295\times0.77\times \]
\[ \times1.05)+(0.15\times0.295\times0.69\times1.07)=0.0270. \]

The percentage content of manganese can be obtained by substituting the value found into the formula

\[ \% \text{ manganese}= \frac{\dfrac{C^\beta}{Y_{\text{total}}}\cdot M\cdot 100} {6.02\cdot10^{23}\cdot \text{grams of alloy}\cdot n\nu_{\text{act}}(1-e^{-\lambda t})}. \tag{23'} \]

Before completing the numerical calculation outlined above, it is necessary to make one comment concerning the proper use of the activation cross section \(\sigma_{\text{act}}\) taken from Table I. The value \(13\cdot10^{-24}\) cm\(^2\)/atom given above refers to the neutron velocity \(v_0=2200\) m/sec, which corresponds to a neutron energy of 0.025 eV. The energies of pile neutrons, however, are distributed over a broad spectrum. This is clear from the following. First, approximately 5% of the activated manganese atoms irradiated at

end of the pneumatic tube, results from the capture of fast neutrons. This can be shown by comparing the activity of two identical samples, one of which, during irradiation, was wrapped in cadmium foil 1 mm thick. Thus, in order to take account of the dependence of activation cross sections according to the law \(1/v\), the total number of pulses must additionally be multiplied by 0.95. Secondly, although the thermal part of the neutron spectrum of the pile does obey the Maxwellian velocity distribution, the mean velocity nevertheless somewhat exceeds \(2200\ \text{m/sec}\) (\(300^\circ\) K). The standard value \(\sigma_{\mathrm{ak}}\) for \(\mathrm{Mn}^{56}\) must therefore be increased proportionally. In general, to find the mean cross section \(\bar{\sigma}\) of an absorber obeying the \(1/v\) law under a Maxwellian distribution, the values in Table I should be multiplied by

\[ \frac{1}{1.128}\sqrt{\frac{300}{T}}. \]

In the case under consideration \(\bar{\sigma}_{\mathrm{ak}}=0.658\sigma_{\mathrm{ak}}=8.5\cdot 10^{-24}\). Finally we obtain:

\[ \% \text{ manganese} = \frac{\left(\dfrac{159.5}{2.7}\cdot 10^{-2}\right)\cdot 55\cdot 100} {6.02\cdot 10^{23}\cdot 2.5\cdot 10^{-2}\cdot 6.4\cdot 10^{11}\cdot 8.5\cdot 10^{-24}\cdot 2.22\cdot 10^{-3}} =1.69. \]

Chemical analysis of the same alloy gave the value 1.48%. In view of a number of uncertain factors and a number of known experimental errors in absolute measurements, it must be admitted that the agreement of the two results is clearly within the limits of the experimental errors that occurred in the determination by the activation method. A considerably more accurate determination of the amount of manganese could have been obtained by the comparison method.

The method of activation analysis can be used to determine the isotopic content of stable isotopes, provided that satisfactory half-lives are obtained. Moreover, by means of the comparative method it will be possible to measure changes in isotopic composition with an accuracy not inferior to that of modern mass spectrometry. This method has already been used to check the degree of separation of chlorine isotopes in a thermal diffusion column[^18]. As an illustration, let us present data from preliminary experiments on determining the percentage ratio of copper isotopes[^*].

The isotopic composition was determined of a preparation of copper isotopes made in a calutron. Analysis by the mass-spectrographic method gave values of \(97\pm 1\%\) for the isotope of mass 63 and \(3\pm 1\%\) for mass 65. Part of this preparation, weighing 0.026 mg, was irradiated in the pile for 5 minutes, with the formation of \(\mathrm{Cu}^{64}\) (12.8 hours) and \(\mathrm{Cu}^{66}\) (5 min.). The total decay curve was taken and was then resolved with maximum accuracy into its two components. Extrapolating to zero time, it was possible to assign

\(4 \cdot 10^4\) β-pulses per minute over a five-minute period at a geometric factor of 30%. Using these data, as well as data on the magnitude of the neutron flux and on the isotopic cross section for \(\mathrm{Cu}^{65}\), it was found that the amount of isotope 65 in the sample was \(9.3 \cdot 10^{-7}\) g. Hence a value of 3.5% \(\mathrm{Cu}^{65}\) follows, which is in better agreement with the result of mass-spectrographic analysis than could have been expected, taking into account the numerous approximations and errors inevitable in the activation method.

CONCLUSIONS

Our aim has been to examine the present state of the activation-analysis method and to illustrate its application by various examples. As an analytical tool this method is, undoubtedly, only at the initial stage of its development, although the establishment of the physical foundations necessary for its broad application is rapidly being completed. Up to now the new method has most often been applied for purposes of analyzing metals and alloys. In practice, the advisability of using it for solving any given analytical problem must be decided by preliminary consideration of the corresponding nuclear data. The methods of nuclear activation, the choice of the most suitable instrument for measuring radioactivity, and the necessity of chemical separation are factors of particular importance. Since short-lived activities make it possible to obtain the greatest sensitivity, and chemical separations are, as a rule, desirable, the success of the new method in the future will possibly depend on the development of rapid-separation techniques. This problem is at present one of the principal ones in the field of radiochemistry of nuclear-fission products. The author’s experience shows that separations based on differences in volatility or on selective extraction can in some cases be carried out rapidly and with high efficiency.

In institutions not engaged specifically with problems of atomic energy, the broad application of the activation method for determining microcomposition will depend to a considerable extent on the availability of inexpensive installations for isotope production. It would seem that a neutron source with an intensity of \(10^{10}\) neutrons/\(\mathrm{cm}^2\) sec would be quite satisfactory. Because of the poor irradiation geometry, however, a source approximately 50 times more intense will be required. A low-voltage linear accelerator or a small compact cyclotron satisfying these requirements and providing beams of the order of 1 ma (1000 µa) is entirely feasible. Such an installation could be used both for bombardment with charged particles and for obtaining fast and slow neutrons.

Finally, mention should also be made of the possibility of continuous mechanized analysis. If, for example, it were possible to pass samples along a conveyor first for irradiation and then under a detector to measure their activity, we would obviously obtain a convenient method of industrial control.

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